Chaotic Mixing in a Twisted Pipe: Optimisation of Heat, Mass Transfer and RTD

It is now well-known that chaotic advection in laminar flow significantly impacts scalar transport – heat, mass and residence time distribution (RTD). In context of twisted pipes, such phenomena are relevant to a wide range of applications which demand rapid heat and mass transport ranging from microfluidics and continuous chemical processing to bioreactors. A general theoretical framework linking the three modes of transport was described and applied to optimise the performance of a twisted pipe in terms of mixing, heat transfer and ability to produce a narrow RTD. The work indicates that it is possible to optimise all three aspects of the twisted pipe in a single analysis. Introduction Chaotic advection in laminar flow provides a transport mechanism for scalars such as heat and mass with similar characteristics to that of turbulent flow [3, 5]. Mass and heat transfer may be readily accelerated, whilst RTD variance is suppressed, all of which are highly beneficial for a wide range of engineering applications in the laminar flow regime. In flow chemistry, for example, accelerated heat and mass transport respectively are critical for improving mixing and achieving better control of process temperature, whilst product uniformity is directly related to narrowness of the RTD. As many engineering processes require all three modes to be simultaneously optimised, there exists a direct need to understand how this might be achieved in chaotic flows. There exists a deep connection between the three transport modes (heat, mass, RTD) which points to a new methodology for simultaneous optimisation. In this study, we use the twisted pipe flow [3, 12, 25] as a prototypical example to both demonstrate the underlying concepts and perform an optimisation process. We consider how the eigenmode structure in a twisted pipe controls both transverse and axial scalar dispersion, leading to a framework under which global optimisation is possible. Background The twisted pipe flow comprises of a series of pipe bends connected in series with an angular offset (i.e. twist) between two consecutive bends (Figure 1). At certain Reynolds number, fluid passing through any bend induces a secondary flow that arises as the faster fluid stream near the pipe axis moves towards the concave side of the bend [6], resulting in the well-known formation of counter rotating vortices [7, 8] known classically as Dean roll cells. This produces periodic re-orientation of the flow which is critical for sustaining chaotic advection in the pipe. Jones et al. [12] found that the twist angle has considerable effect on transverse scalar transport, specifically that such particular twist angles can generate chaotic mixing within the twisted pipe flow. The twisted pipe flow falls into a wide class of periodically reoriented duct flows, e.g. RAM [15], SMW mixer [23], etc, which has received widespread attention. Various other studies have looked into different twisted pipe configurations to optimise chaotic mixing and heat transfer, for example [1, 18, 19, 25]. At certain twist angles, reorientation of the secondary flow repetitively stretches and folds the trajectories of fluid particles, producing exponential separation of neighbouring fluid particles and highly striated material distributions. This process is the hallmark of chaotic advection, and leads to highly efficient mixing. In conjunction with chaotic advection, thermal or molecular diffusion transport acts to impart significantly accelerated irreversible dispersion, leading to rapid heat and mass transfer. It is known that enhanced transverse dispersion and hence mixing suppresses axial dispersion [13], leading to a narrower RTD [3, 21]. This suggests a deep connection between chaotic mixing and RTD as demonstrated indirectly by Mezić et al. [17] who related RTD to Poincaré map in pipes. Theory Twisted pipe flow is described as a periodically reoriented duct flow with axial coordinate z which aligns with the direction of the bulk flow, and transverse coordinates (r,θ). The 3D periodic velocity field u(x) can be written as , , + = , , (1) where Rθ is a rotation operator about the z-axes, and x = r,θ,z forms an orthogonal coordinate system. Transport of scalars such as heat and mass in this flow can be described by the dimensionless steady advection-diffusion equation (ADE) for the scalar quantity φ representing heat or mass concentration ∙ =  + (2) where Pe is Peclet number for heat scalar ADE, and is replaced with Schmidt number (Sc) for mass scalar AED; S is a domain source. The ADE (2) is subject to the initial condition φ(r,θ,0) = φ0, and either Dirichlet or Neumann boundary conditions respectively at the pipe wall: | = (3) = (4) The Dirichlet boundary condition (3) corresponds to problems with a fixed value at wall, e.g. wall heating. The Neumann boundary condition (4) represents zero flux condition at the wall, e.g. mass transport and RTD evolution. The probability density function (PDF) P of the residence time is given by an unsteady ADE [24] ∙ ! =  ! − # (5) with s being age of the fluid particles. This is remarkably similar to equation (2). It demonstrates a close connection between axial/transverse dispersion and RTD similar to that exists for Taylor-Aris dispersion. Based on Liu and Haller [16], Lester et al. [14] established that, for a spatially periodic flow where u(z) = u(z + L), the solution to the ADE with S(x) = f1(x) = f2(x) = 0 is the sum of a finite number of strange eigenmodes (so-called as the eigenmodes have non-trivial structure in the limit Pe→∞) , , =< % > +'()*+,-./) 0